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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>(<dfn class="terminology">Case 3</dfn>) <span class="process-math">\(g(x)=e^{\alpha x} \cos \beta x P_m(x)\text{.}\)</span>(3 a) <span class="process-math">\(\alpha+i \beta\)</span> is not a root of the characteristic equation, then the particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=e^{\alpha x} \cos \beta x (A_0 x^m+A_1 x^{m-1}+\cdots+A_m)+
e^{\alpha x} \sin \beta x (B_0 x^m+B_1 x^{m-1}+\cdots+B_m).
\end{equation*}
</div>
<p class="continuation">(3 b) <span class="process-math">\(\alpha+i \beta\)</span> is a <span class="process-math">\(s\)</span>-time repeated root, then the particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=x^s e^{\alpha x} \cos \beta x (A_0 x^m+A_1 x^{m-1}+\cdots+A_m)+
x^s \sin \beta x (B_0 x^m+B_1 x^{m-1}+\cdots+B_m).
\end{equation*}
</div>
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